Minimum-Length Coverage Path Planning for Grid Environments With Approximation Guarantees
Bibliographic record
Abstract
We focus on planning minimum-length robot paths to cover environments using the robot's sensor or coverage (e.g., cleaning) tool. Many algorithms use the following framework: (i) compute a grid decomposition of the environment, (ii) partition the grid to be covered by non-overlapping <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">coverage lines</i> (straight-line paths), and (iii) compute a cost-minimizing tour of the coverage lines to get a coverage path. While this framework aims to minimize turns in the path, it does not yield guarantees on the resulting path length. In this paper, we show that this framework guarantees a coverage path of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1 + 1.5\gamma)$</tex-math></inline-formula> times the optimal, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma \gt 1$</tex-math></inline-formula> is the approximation factor to solve the metric traveling salesman problem (metric-TSP). Following this, we propose the Minimum Length Coverage Approx (MLC-Approx) approach that modifies this framework to achieve an approximation factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1.5 + \epsilon)$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon \ll 1$</tex-math></inline-formula> depends on the number of coverage lines. Instead of computing a tour of the coverage lines, MLC-Approx merges minimum-length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sub-tours</i> of coverage lines while minimizing the turns added by the merges. We also propose a lazy variation of MLC-Approx that achieves the same result with faster empirical runtime. We validate MLC-Approx in simulations using maps of real-world environments and compare against state-of-the-art CPP approaches.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".